The invention described in this patent application relates to the problem of estimation of constant parameters of multiple signals received by an array of sensors in the presence of additive noise. There are many physical problems of this type including direction finding (DF) wherein the signal parameters of interest are the directions-of-arrival (DOA's) of wavefronts impinging on an antenna array (cf. FIG. 1), and harmonic analysis in which the parameters of interest are the temporal frequencies of sinusoids contained in a signal (waveform) which is known to be composed of a sum of multiple sinusoids and possibly additive measurement noise. In most situations, the signals are characterized by several unknown parameters all of which need to be estimated simultaneously (e.g., azimuthal angle, elevation angle and temporal frequency) and this leads to a multidimensional parameter estimation problem.
High resolution parameter estimation is important in many applications including electromagnetic and acoustic sensor systems (e.g., radar, sonar, electronic surveillance systems, and radio astronomy), vibration analysis, medical imaging, geophysics, well-logging, etc.. In such applications, accurate estimates of the parameters of interest are required with a minimum of computation and storage requirements. The value of any technique for obtaining parameter estimates is strongly dependent upon the accuracy of the estimates. The invention described herein yields accurate estimates while overcoming the practical difficulties encountered by present methods such as the need for detailed a priori knowledge of the sensor array geometry and element characteristics. The technique also yields a dramatic decrease in the computational and storage requirements.
The history of estimation of signal parameters can be traced back at least two centuries to Gaspard Riche, Baron de Prony, (R. Prony, Essai experimental et analytic, etc. L'Ecole Polytechnique, 1: 24-76, 1795) who was interested in fitting multiple sinuisoids (exponentials) to data. Interest in the problem increased rapidly after World War II due to its applications to the fast emerging technologies of radar, sonar and seismology. Over the years, numerous papers and books addressing this subject have been published, especially in the context of direction finding in passive sensor arrays.
One of the earliest approaches to the problem of direction finding is now commonly referred to as the conventional beamforming technique. It uses a type of matched filtering to generate spectral plots whose peaks provide the parameter estimates. In the presence of multiple sources, conventional beamforming can lead to signal suppression, poor resolution, and biased parameter (DOA) estimates.
The first high resolution method to improve upon conventional beamforming was presented by Burg (J. P. Burg, Maximum entropy spectral analysis, In Proceedings of the 37.sup.th Annual International SEG Meeting, Oklahoma City, OK., 1967). He proposed to extrapolate the array covariance function beyond the few measured bags, selecting that extrapolation for which the entropy of the signal is maximized. The Burg technique gives good resolution but suffers from parameter bias and the phenomenon referred to as line splitting wherein a single source manifests itself as a pair of closely spaced peaks in the spectrum. These problems are attributable to the mismodeling inherent in this method.
A different approach aimed at providing increased parameter resolution was introduced by Capon (J. Capon, High resolution frequency wave number spectrum analysis, Proc. IEEE, 57: 1408-1418, 1969). His approach was to find a weight vector for combining the outputs of all the sensor elements that minimizes output power for each look direction while maintaining a unit response to signals arriving from this direction. Capon's method has difficulty in multipath environments and offers only limited improvements in resolution.
A new genre of methods were introduced by Pisarenko (V. F. Pisarenko, The retrieval of harmonics from a covariance function, Geophys. J. Royal Astronomical Soc., 33: 347-366, 1973) for a somewhat restricted formulation of the problem. These methods exploit the eigenstructure of the array covariance matrix. Schmidt made important generalizations of Pisarenko's ideas to arbitrary array/wavefront geometries and source correlations in his Ph.D. thesis titled A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation, Stanford University, 1981. Schmidt's MUltiple SIgnal Classification (MUSIC) algorithm correctly modeled the underlying problem and therefore generated superior estimates. In the ideal situation where measurement noise is absent (or equivalently when an infinite amount of measurements are available), MUSIC yields exact estimates of the parameters and also offers infinite resolution in that multiple signals can be resolved regardless of the proximity of the signal parameters. In the presence of noise and where only a finite number of measurements are available, MUSIC estimates are very nearly unbiased and efficient, and can resolve closely spaced signal parameters.
The MUSIC algorithm, often referred to as the eigenstructure approach, is currently the most promising high resolution parameter estimation method. However, MUSIC and the earlier methods of Burg and Capon which are applicable to arbitrary sensor array configurations suffer from certain shortcomings that have restricted their applicability in several problems. Some of these are:
Array Geometry and Calibration--A complete characterization of the array in terms of the sensor geometry and element characteristics is required. In practice, for complex arrays, this characterization is obtained by a series of experiments known as array calibration to determine the so called array manifold. The cost of array calibration can be quite high and the procedure is sometimes impractical. Also, the associated storage required for the array manifold is 2ml.sup.g words (m is the number of sensors, l is the number of search (grid) points in each dimension, and g is the number of dimensions) and can become large even for simple applications. For example, a sensor array containing 20 elements, searching over a hemisphere with a 1 millirad resolution in azimuth and elevation and using 16 bit words (2 bytes each) requires approximately 100 megabytes of storage! This number increases exponentially as another search dimension such as temporal frequency is included. Furthermore, in certain applications the array geometry may be slowly changing such as in light weight spaceborne antenna structures, sonobuoy and towed arrays used in sonar etc., and a complete characterization of the array is never available.
Computational Load--In the prior methods of Burg, Capon, Schmidt and others, the main computational burden lies in generating a spectral plot whose peaks correspond to the parameter estimates. For example, the number of operations required in the MUSIC algorithm in order to compute the entire spectrum, is approximately 4m.sup.2 l.sup.g. An operation is herein considered to be a floating point multiplication and an addition. In the example above, the number of operations needed is approximately 4.times.10.sup.9 which is prohibitive for most applications. A powerful 10 MIP (10 million floating point instructions per second) machine requires about 7 minutes to perform these computations! Moreover, the computation requirement grows exponentially with dimension of the parameter vector. Augmenting the dimension of the parameter vector further would make such problems completely intractable.
The technique described herein is hereafter referred to as Estimation of Signal Parameters using Rotational Invariance Techniques (ESPRIT). ESPRIT obviates the need for array calibration and dramatically reduces the computational requirements of previous approaches. Furthermore, since the array manifold is not required, the storage requirements are eliminated altogether.